In contrast to the generic situation for deterministic inverse problems, it has been shown that the corresponding Bayesian inverse problem for PDEs is often well-posed i.e., the posterior measures exists, is unique and depends continuously (in suitable metrics) on the measurements (REF ) [1]}, [2]}, [3]}. Furthermore, Bayesian inverse problems can incorporate the deterministic formulation of regularized ill-posed inverse problems: As shown in [1]}, the latter can often be viewed as the maximum a posteriori (MAP) estimator of a Bayesian inverse problem with a suitable choice of the underlying prior.
